Finite Group Databasebeta

← #28401376

Subgroups — #28401376

Order 14464.

462 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813596C211yesmaximal, characteristic
2#20298931C212yesmaximal, complemented
3#20298911C212yesmaximal, complemented
4#20298846C214yesmaximal, complemented
5#20298748C212yesmaximal, complemented
6#20298737C212yesmaximal, complemented
7#20298562C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C113C211yesmaximal, characteristic, complemented
9C22×D452C2211yescharacteristic
10C22×C22⋊D113—21no—
11C2×#14268634C2214yescomplemented
12C2×#14268631C2212yescomplemented
13#16813596—21no—
14#16813596—21no—
15#16813596C2212yes—
16#16813596C2212yes—
17C2×C4⋊Dic113C2211yescharacteristic
18C2×#14268624—21no—
19#16813555C2214yescomplemented
20#16813546C2214yescomplemented
21#16813542C2214yescomplemented
22C22⋊#11744751C2212yes—
23C22⋊#11744751C2212yescomplemented
24C22×Q8×C113C2211yescharacteristic, complemented
25C23×C4×C113C2211yescharacteristic
26C22⋊Q8×C113—21no—
27C22⋊Q8×C113—21no—
28C22⋊Q8×C113C2212yes—
29C22⋊Q8×C113C2212yescomplemented
30C2×C4⋊C4×C113—21no—
31C2×C4⋊C4×C113C2211yescharacteristic, complemented
32C2×C22⋊C4×C113—21no—
33C23×D113—21no—
34C2×D452—21no—
35C2×D452—42no—
36C2×D452C2312yes—
37C2×C22⋊D113—22no—
38C2×C22⋊D113—22no—
39C2×C22⋊D113—44no—
40C22×Dic113—21no—
41#14268634—44no—
42#14268631—42no—
43C2×#11744751C2314yes—
44C4⋊Dic113—21no—
45C4⋊Dic113C2312yes—
46#14268624—22no—
47#14268624—22no—
48C24×C113D411yescharacteristic
49C2×Q8×C113—21no—
50C2×Q8×C113—42no—
51C2×Q8×C113C2312yes—
52C22×C4×C113—21no—
53C22×C4×C113—21no—
54C22×C4×C113—42no—
55C22×C4×C113D411yescharacteristic
56C22×C4×C113D411yescharacteristic
57C22×C4×C113D411yescharacteristic
58C22×C4×C113C2311yescharacteristic
59C22×C4×C113C2311yescharacteristic
60C22×C4×C113C2311yescharacteristic
61C4⋊C4×C113—22no—
62C4⋊C4×C113—22no—
63C4⋊C4×C113—21no—
64C4⋊C4×C113C2312yes—
65C22⋊C4×C113—22no—
66C22⋊C4×C113—22no—
67C23×C113—22no—
68C23×C113—22no—
69C23×C113—41no—
70C23×C113C2×D411yescharacteristic
71C23×C113C2×D411yescharacteristic
72C23×C113C2×D411yescharacteristic
73C22×D113—22no—
74C22×D113—42no—
75C22×D113—42no—
76C22×D113—42no—
77Q8×C113—24no—
78Q8×C113—42no—
79C2×C4×C113—22no—
80C2×C4×C113—22no—
81C2×C4×C113—22no—
82C2×C4×C113—22no—
83C2×C4×C113—22no—
84C2×C4×C113—41no—
85C2×C4×C113—41no—
86C2×C4×C113—41no—
87C2×C4×C113—42no—
88C2×C4×C113—42no—
89C2×C4×C113C2×D411yescharacteristic
90C2×C4×C113C2×D411yescharacteristic
91C2×C4×C113C2×D411yescharacteristic
92C2×C4×C113C2411yescharacteristic
93C22⋊D113—44no—
94C22⋊D113—44no—
95C2×Dic113—22no—
96C2×Dic113—42no—
97D452—24no—
98D452—42no—
99#11744751—24no—
100C22×C113—21no—
101C22×C113—21no—
102C22×C113—21no—
103C22×C113—21no—
104C22×C113—21no—
105C22×C113—21no—
106C22×C113—21no—
107C22×C113—44no—
108C22×C113AGL(1,ℤ/8)12yes—
109C22×C113#13712yes—
110C22×C113C22×D411yescharacteristic
111C2×D113—42no—
112C2×D113—42no—
113C2×D113—42no—
114C2×D113—84no—
115C4×C113—21no—
116C4×C113—21no—
117C4×C113—21no—
118C4×C113—42no—
119C4×C113—42no—
120C4×C113H(5,𝔽2)12yes—
121Dic113—42no—
122C2×C113—22no—
123C2×C113—22no—
124C2×C113—41no—
125C2×C113#53411yescharacteristic
126C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C113C2×#13711yescharacteristic
128D113—82no—
129#3193—1131nomaximal, Sylow
130C113#319311yescharacteristic, complemented, Sylow
131C2×C22⋊Q8D11311yescharacteristic, complemented
132C2×#121—1131no—
133#484—1132no—
134#480—1132no—
135#468—1134no—
136#450—1132no—
137#448—1132no—
138#407—1131no—
139C22×Q8C2×D11311yescharacteristic, complemented
140C22×D4—1131no—
141C22×D4—2261no—
142C23×C4C2×D11311yescharacteristic
143C2×SD16—1134no—
144C2×M16—1132no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8C2×D11312yes—
148C22⋊Q8C2×D11312yescomplemented
149#121—1132no—
150#121—1132no—
151#121—2261no—
152#121—2261no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—1131no—
155C2×C4⋊C4C2×D11311yescharacteristic, complemented
156C2×C22⋊C4—21no—
157C2×C22⋊C4—2261no—
158#107—1134no—
159#103—1134no—
160#102—1134no—
161C22⋊C8—1132no—
162C22⋊C8—1132no—
163C24—2261no—
164C24C22⋊D11311yescharacteristic
165C2×Q8—21no—
166C2×Q8—42no—
167C2×Q8C22×D11312yes—
168C2×D4—1132no—
169C2×D4—4522no—
170C2×D4—4524no—
171C2×D4—2262no—
172C2×D4—2262no—
173C2×D4—2261no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—2261no—
178C22×C4C22⋊D11311yescharacteristic
179C22×C4C22⋊D11311yescharacteristic
180C22×C4C22⋊D11311yescharacteristic
181C22×C4C22×D11311yescharacteristic
182C22×C4C22×D11311yescharacteristic
183C22×C4C22×D11311yescharacteristic
184SD16—4524no—
185M16—4522no—
186C2×C8—1134no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—1132no—
191C4⋊C4—2261no—
192C4⋊C4C22×D11312yes—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2262no—
196C22⋊C4—2262no—
197C23—22no—
198C23—22no—
199C23—41no—
200C23—4522no—
201C23—4522no—
202C23—4522no—
203C23—2262no—
204C23C2×C22⋊D11311yescharacteristic
205C23C2×C22⋊D11311yescharacteristic
206C23C2×C22⋊D11311yescharacteristic
207Q8—24no—
208Q8—42no—
209D4—4522no—
210D4—4524no—
211D4—4524no—
212D4—2264no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—4522no—
224C2×C4—2262no—
225C2×C4C2×C22⋊D11311yescharacteristic
226C2×C4C2×C22⋊D11311yescharacteristic
227C2×C4C2×C22⋊D11311yescharacteristic
228C2×C4C23×D11311yescharacteristic
229C8—2264no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—44no—
238C22—9044no—
239C22—4522no—
240C22—4522no—
241C22—4522no—
242C22#1681364712yes—
243C22#1681365312yes—
244C22C22×C22⋊D11311yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—4522no—
251C4#1681366712yes—
252C2—22no—
253C2—22no—
254C2—41no—
255C2—9042no—
256C2#2029915711yescharacteristic
257C2C2×#1681364711yescharacteristic
258C2C2×#1681365311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×C22⋊Q8×C113C2
socleC22×C113C22×D4
prime radicalC2×#16813596C2
hypercenterC2×C22⋊Q8D113
hypocenterC113#3193
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20298931C2semidirect
3. #20298911C2semidirect
4. #20298846C2semidirect
5. #20298748C2semidirect
6. #20298737C2semidirect
7. #20298562C2semidirect
8. C2×C22⋊Q8×C113C2semidirect
11. C2×#14268634C22semidirect
12. C2×#14268631C22semidirect
19. #16813555C22semidirect
20. #16813546C22semidirect
21. #16813542C22semidirect
23. C22⋊#11744751C22semidirect
24. C22×Q8×C113C22semidirect
29. C22⋊Q8×C113C22semidirect
31. C2×C4⋊C4×C113C22semidirect
130. C113#3193semidirect
131. C2×C22⋊Q8D113semidirect
139. C22×Q8C2×D113semidirect
148. C22⋊Q8C2×D113semidirect
155. C2×C4⋊C4C2×D113semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168135962
#202989312
#202989112
#202988462
#202987482
#202987372
#202985622
C2×C22⋊Q8×C1132
#3193113